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ISBN 10: 1154974839 ISBN 13: 9781154974836

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**Item Description: **RareBooksClub. Paperback. Book Condition: New. This item is printed on demand. Paperback. 94 pages. This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1853 edition. Excerpt: . . . plane. This method may serve to verify the calculations which are made by the rules in arts. 50 and 51. Ex. If one end of the frustum CDGL (Fig. 17. ) be 90 feet square, the other end 60 feet square, and the height RN 36 feet; what is the height of the whole pyramid VCD: and what are the solidity and lateral surface of the frustum DPDN--GR45--3015. And, GPRN-36. Then, 15: 36: : 45: 108VN, the height of the whole pyramid. And, 108--3672VR, the height of the part VLG. The solidity of the large pyramid is 291600 (Art. 48. ) of the small pyramid 86400 of the frustum CDGL 2(15200 The lateral surface of the large pyramid is 21060 (Art. 49. ) of the small pyramid 9360 of the frustum 11700 PROBLEM VII. To find the Solidity of a Wedge. 54. Add The Length Of The Edge To Twice The Length Of The Base, And Multiply The Sum Bv J Of The Product Of The Height Of The Wedge And The Breadth Of The Base. Let LAB the length of the base. (Fig. 20. ) GH the length of the edge. 6 BC the breadth of the base. APG the height of the wedge. Then, L--lAB--GHAM. If the length of the base and the edge be equal, as BM and GH, (Fig. 20. ) the wedge MBHG is half a parallelopiped of the same base and height. And the solidity (Art. 43. ) is equal to half the product of the height, into the length and breadth of the base; that is to I bhl. If the length of the base be greater than that of the edge, as ABGH; let a section be made by the plane GMN, parallel to HBC. This will divide the whole wedge into two parts MBHG and AMG. The latter is a pyramid, whose solidity (Art. 48. ) is iMx(L--I) The solidity of the parts together, is, therefore, i bhll bh x(L--)-bh3l6A2L--bh 2libhx(2Ll) If the length of the base be less than that of the edge, it is evident that the pyramid. . . This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781154974836

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**Item Description: **2015. Paperback. Book Condition: New. Reprinted from 1836 edition. NO changes have been made to the original text. Each page is checked manually before printing. Illustrations, Index, if any, are included in b/w. Fold-outs, if any, are not part of the book. If the original book was published in multiple volumes than this reprint is of only one volume. This paperback book is SEWN perfect bound, where the book block is actually sewn (smythe sewn/section sewn) with thread before binding which results in a more durable type of paperback binding. It can also be open wide. The pages will not fall out and will be around for a lot longer than normal paperbacks. This book is printed on demand on acid-free paper.(Original publisher, H. Howe & co.) 407 pages Special Offer: This seller is offering an exclusive discount of 10% on all prices for AbeBooks customers. Original Price: 28.85 EUR. Bookseller Inventory # 452460129

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**Item Description: **2015. Paperback. Book Condition: New. Reprinted from 1831 edition. NO changes have been made to the original text. Each page is checked manually before printing. Illustrations, Index, if any, are included in b/w. Fold-outs, if any, are not part of the book. If the original book was published in multiple volumes than this reprint is of only one volume. This paperback book is SEWN perfect bound, where the book block is actually sewn (smythe sewn/section sewn) with thread before binding which results in a more durable type of paperback binding. It can also be open wide. The pages will not fall out and will be around for a lot longer than normal paperbacks. This book is printed on demand on acid-free paper.(Original publisher, H. Howe) 437 pages Special Offer: This seller is offering an exclusive discount of 10% on all prices for AbeBooks customers. Original Price: 30.35 EUR. Bookseller Inventory # 452486716

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Published by Nabu Press

ISBN 10: 1141258676 ISBN 13: 9781141258673

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**Item Description: **Nabu Press. Paperback. Book Condition: New. This item is printed on demand. Paperback. 286 pages. Dimensions: 9.7in. x 7.4in. x 0.6in.This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781141258673

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Published by Rarebooksclub.com, United States (2012)

ISBN 10: 1231205318 ISBN 13: 9781231205310

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**Item Description: **Rarebooksclub.com, United States, 2012. Paperback. Book Condition: New. 246 x 189 mm. Language: English Brand New Book ***** Print on Demand *****. This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1854 Excerpt: .the two ends, there may be drawn in each, such a number of small rectangles, that the sum of them shall differ less, than by any given quantity, from the figure in which they are contained. And the solids between these rectangles will be rectangular prismoids. Ex. 1. If one end of a rectangular prismoid be 44 feet by 23, the other end 36 by 21, and the perpendicular height 72; what is the solidity? The area of the larger end =44x23=1012 of the smaller end =36x21= 756 of the middle section =40x22= 880 And the solidity =(1012+756 +4x880)xl2=63456 feet. 2. What is the solidity of a stick of hewn timber, whose ends are 30 inches by 27, and 24 by 18, and whose length is 48 feet? Ans. 204 feet. Other solids not treated of in this section, if they be bounded by plane surfaces, may be measured by supposing them to be divided into prisms, pyramids, and wedges. And, in.v deed, every such sobi may be considered as made up of triangular pyramids. THE FIVE REGULAR SOLIDS. 56. A SOLID IS SAID TO BE REGULAR, WHEN ALL ITS SOLID ANGLES ARE EQUAL, AND ALL ITS SIDES ARE EQUAL AND REGULAR POLYGONS. The following figures are of this description; 1. The Tetraedron, 2. The Hexaedron or cube, 3. The Octaedron, 4. The Dodecaedron, 5. The Icosaedron, whose sides are four triangles; six squares; eight triangles; twelve pentagons; twenty triangles. Besides these five, there can be no other regular solids. The only plane figures which can form such solids, are triangles, squares, and pentagons. For the plane angles which contain any solid angle, are together. less than four right angles or 360. (Sup. Euc. 21,2.) And the least number which can form a solid angle is three. (Sup. Euc. Def. 8, 2.) If they are angles of equilateral triangles, each is 60. The sum of three of them is 180, of . Bookseller Inventory # APC9781231205310

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Published by Rarebooksclub.com, United States (2012)

ISBN 10: 1130343022 ISBN 13: 9781130343021

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**Item Description: **Rarebooksclub.com, United States, 2012. Paperback. Book Condition: New. 246 x 189 mm. Language: English Brand New Book ***** Print on Demand *****. This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1838 Excerpt: .of the base; and therefore, the sum of the breadths is equal to the perimeter of the base. Ex. 1. If the base of a right prism be a regular hexagon whose sides are each 2 feet 3 inches, and if the height be 16 feet, what is the lateral surface Ans. 216 square feet. If the areas of the two ends be added to the lateral surface, the sum will be the whole surface of the prism. And the superficies of any solid bounded by planes, is evidently equal to the areas of all its sides. Ex. 2. If the base of a prism be an equilateral triangle whose perimeter is 6 feet, and if the height be 17 feet, what is the surface 1 The area of the triangle is 1.732. (Art. 11.) And the whole surface is 105.464. PROBLEM III. To find the Solidity of a Pyramid. 48. Multiply The Area Of The Base Into i Of The Height. The solidity of a prism is equal to the product of the area of the base into the height. (Art. 43.) And a pyramid is i of a prism of the same base and altitude. (Sup. Euc. 15, 3. Cor. 1.) Therefore the solidity of a pyramid whether right or oblique, is equal to the product of the base into A of the perpendicular height. Ex. 1. What is the solidity of a triangular pyramid, whose height is 60, and each side of whose base is 41 The area of the base is 6.928 And the solidity is 138.56. 2. Let ABC (Fig. 16.) be one side of an oblique pyramid whose base is 6 feet square; let BC be 20 feet, and make an angle of 70 degrees with the plane of the base; and let CP be perpendicular to this plane. What is the.solidity of the pyramid? 3. What is the lateral surface of a regular pyramid whose axis is 12 feet, and whose base is 18 feet square? Ans. 540 square feet. The lateral surface of an oblique pyramid may be found, by taking the sum of the areas of the unequal triangles which form it. Bookseller Inventory # APC9781130343021

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Published by Rarebooksclub.com, United States (2012)

ISBN 10: 1130655547 ISBN 13: 9781130655544

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**Item Description: **Rarebooksclub.com, United States, 2012. Paperback. Book Condition: New. 246 x 189 mm. Language: English Brand New Book ***** Print on Demand *****. This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1831 Excerpt: .Therefore the area of all the equal trapezoids which form the lateral surface of Sea note F. the frustum, is equal to the product of half the slant-height into the sum of the perimeters of the ends. Ex. If the slant-height of a frustum of a regular octagonal pyramid be 42 feet, the sides of one end 5 feet each, and the sides of the other end 3 feet each; what is the lateral surface?, Ans. 1344 square feet. 52. If the slant-height be not given, it may be obtained from the perpendicular height, and the dimensions of the two ends. Let GD (Fig. 17.) be the slant-height of the frustum CDGL, RN or GP the perpendicular height, ND and RG the radii of the circles inscribed in the perimeters of the two ends. Then PD is the difference of the two radii: And the slant-height GD = x/(GP3+YD3). Ex. If the perpendicular height of a frustum of a regular hexagonal pyramid be 24, the sides of one end 13 each, and the sides of the other end 8 each; what is the whole surface? t/(BC-BP)=CP, (Fig.7.)thati8, .v/(l33-633)=11.258 And V 83--43 e= 6.928 The difference of the two radii is, therefore, 4.33 The slant-height =V(243 + 4.333)=24.3875 The lateral surface te 1536.4 And the whole surface, 2141.75. 53. The height of the whole pyramid may be calculated from the dimensions of the frustum. Let VN (Fig. 17.) be the height of the pyramid, RN or GP the height of the frustum, ND and RG the radii of the circles inscribed in the perimeters of the ends of the frustum. Then, in the similar triangles GPD and VND, DP: GP:: DN: VN. The height of the frustum subtracted from VN, gives VR the height of the small pyramid VLG. The solidity and lateral surface of the frustum may then be found, by subtracting from the whole pyramid, the part which is above the cut ting plane. This method may serve to . Bookseller Inventory # APC9781130655544

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ISBN 10: 1231205318 ISBN 13: 9781231205310

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**Item Description: **RareBooksClub. Paperback. Book Condition: New. This item is printed on demand. Paperback. 90 pages. Dimensions: 9.7in. x 7.4in. x 0.2in.This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1854 Excerpt: . . . the two ends, there may be drawn in each, such a number of small rectangles, that the sum of them shall differ less, than by any given quantity, from the figure in which they are contained. And the solids between these rectangles will be rectangular prismoids. Ex. 1. If one end of a rectangular prismoid be 44 feet by 23, the other end 36 by 21, and the perpendicular height 72; what is the solidity The area of the larger end 44x231012 of the smaller end 36x21 756 of the middle section 40x22 880 And the solidity (1012756 4x880)xl263456 feet. 2. What is the solidity of a stick of hewn timber, whose ends are 30 inches by 27, and 24 by 18, and whose length is 48 feet Ans. 204 feet. Other solids not treated of in this section, if they be bounded by plane surfaces, may be measured by supposing them to be divided into prisms, pyramids, and wedges. And, in. v deed, every such sobi may be considered as made up of triangular pyramids. THE FIVE REGULAR SOLIDS. 56. A SOLID IS SAID TO BE REGULAR, WHEN ALL ITS SOLID ANGLES ARE EQUAL, AND ALL ITS SIDES ARE EQUAL AND REGULAR POLYGONS. The following figures are of this description; 1. The Tetraedron, 2. The Hexaedron or cube, 3. The Octaedron, 4. The Dodecaedron, 5. The Icosaedron, whose sides are four triangles; six squares; eight triangles; twelve pentagons; twenty triangles. Besides these five, there can be no other regular solids. The only plane figures which can form such solids, are triangles, squares, and pentagons. For the plane angles which contain any solid angle, are together. less than four right angles or 360. (Sup. Euc. 21, 2. ) And the least number which can form a solid angle is three. (Sup. Euc. Def. 8, 2. ) If they are angles of equilateral triangles, each is 60. The sum of three of them is 180, of . . . This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781231205310

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ISBN 10: 1130343022 ISBN 13: 9781130343021

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**Item Description: **RareBooksClub. Paperback. Book Condition: New. This item is printed on demand. Paperback. 82 pages. Dimensions: 9.7in. x 7.4in. x 0.2in.This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1838 Excerpt: . . . of the base; and therefore, the sum of the breadths is equal to the perimeter of the base. Ex. 1. If the base of a right prism be a regular hexagon whose sides are each 2 feet 3 inches, and if the height be 16 feet, what is the lateral surface Ans. 216 square feet. If the areas of the two ends be added to the lateral surface, the sum will be the whole surface of the prism. And the superficies of any solid bounded by planes, is evidently equal to the areas of all its sides. Ex. 2. If the base of a prism be an equilateral triangle whose perimeter is 6 feet, and if the height be 17 feet, what is the surface 1 The area of the triangle is 1. 732. (Art. 11. ) And the whole surface is 105. 464. PROBLEM III. To find the Solidity of a Pyramid. 48. Multiply The Area Of The Base Into i Of The Height. The solidity of a prism is equal to the product of the area of the base into the height. (Art. 43. ) And a pyramid is i of a prism of the same base and altitude. (Sup. Euc. 15, 3. Cor. 1. ) Therefore the solidity of a pyramid whether right or oblique, is equal to the product of the base into A of the perpendicular height. Ex. 1. What is the solidity of a triangular pyramid, whose height is 60, and each side of whose base is 41 The area of the base is 6. 928 And the solidity is 138. 56. 2. Let ABC (Fig. 16. ) be one side of an oblique pyramid whose base is 6 feet square; let BC be 20 feet, and make an angle of 70 degrees with the plane of the base; and let CP be perpendicular to this plane. What is the. solidity of the pyramid 3. What is the lateral surface of a regular pyramid whose axis is 12 feet, and whose base is 18 feet square Ans. 540 square feet. The lateral surface of an oblique pyramid may be found, by taking the sum of the areas of the unequal triangles which form it. . . This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781130343021

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Published by RareBooksClub

ISBN 10: 1130655547 ISBN 13: 9781130655544

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**Item Description: **RareBooksClub. Paperback. Book Condition: New. This item is printed on demand. Paperback. 96 pages. Dimensions: 9.7in. x 7.4in. x 0.2in.This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1831 Excerpt: . . . Therefore the area of all the equal trapezoids which form the lateral surface of Sea note F. the frustum, is equal to the product of half the slant-height into the sum of the perimeters of the ends. Ex. If the slant-height of a frustum of a regular octagonal pyramid be 42 feet, the sides of one end 5 feet each, and the sides of the other end 3 feet each; what is the lateral surface, Ans. 1344 square feet. 52. If the slant-height be not given, it may be obtained from the perpendicular height, and the dimensions of the two ends. Let GD (Fig. 17. ) be the slant-height of the frustum CDGL, RN or GP the perpendicular height, ND and RG the radii of the circles inscribed in the perimeters of the two ends. Then PD is the difference of the two radii: And the slant-height GD x(GP3YD3). Ex. If the perpendicular height of a frustum of a regular hexagonal pyramid be 24, the sides of one end 13 each, and the sides of the other end 8 each; what is the whole surface t(BC-BP)CP, (Fig. 7. )thati8, . v(l33-633)11. 258 And V83--43 e 6. 928 The difference of the two radii is, therefore, 4. 33 The slant-height V(243 4. 333)24. 3875 The lateral surface te 1536. 4 And the whole surface, 2141. 75. 53. The height of the whole pyramid may be calculated from the dimensions of the frustum. Let VN (Fig. 17. ) be the height of the pyramid, RN or GP the height of the frustum, ND and RG the radii of the circles inscribed in the perimeters of the ends of the frustum. Then, in the similar triangles GPD and VND, DP: GP: : DN: VN. The height of the frustum subtracted from VN, gives VR the height of the small pyramid VLG. The solidity and lateral surface of the frustum may then be found, by subtracting from the whole pyramid, the part which is above the cut ting plane. This method may serve to . . . This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781130655544

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Published by Nabu Press

ISBN 10: 1146503601 ISBN 13: 9781146503600

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**Item Description: **Nabu Press. Paperback. Book Condition: New. This item is printed on demand. Paperback. 384 pages. Dimensions: 9.7in. x 7.4in. x 0.8in.This is an EXACT reproduction of a book published before 1923. This IS NOT an OCRd book with strange characters, introduced typographical errors, and jumbled words. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781146503600

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Published by Nabu Press

ISBN 10: 1143275713 ISBN 13: 9781143275715

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**Item Description: **Nabu Press. Paperback. Book Condition: New. This item is printed on demand. Paperback. 404 pages. Dimensions: 9.7in. x 7.4in. x 0.8in.This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781143275715

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Published by Nabu Press

ISBN 10: 1144100046 ISBN 13: 9781144100047

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**Item Description: **Nabu Press. Paperback. Book Condition: New. This item is printed on demand. Paperback. 402 pages. Dimensions: 9.7in. x 7.4in. x 0.8in.This is an EXACT reproduction of a book published before 1923. This IS NOT an OCRd book with strange characters, introduced typographical errors, and jumbled words. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781144100047

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Published by Nabu Press

ISBN 10: 1144573017 ISBN 13: 9781144573018

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**Item Description: **Nabu Press. Paperback. Book Condition: New. This item is printed on demand. Paperback. 414 pages. Dimensions: 9.7in. x 7.4in. x 0.8in.This is an EXACT reproduction of a book published before 1923. This IS NOT an OCRd book with strange characters, introduced typographical errors, and jumbled words. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781144573018

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Published by Nabu Press

ISBN 10: 1148966188 ISBN 13: 9781148966182

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**Item Description: **Nabu Press. Paperback. Book Condition: New. This item is printed on demand. Paperback. 418 pages. Dimensions: 9.7in. x 7.4in. x 0.8in.This is an EXACT reproduction of a book published before 1923. This IS NOT an OCRd book with strange characters, introduced typographical errors, and jumbled words. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781148966182

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ISBN 10: 114201780X ISBN 13: 9781142017804

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**Item Description: **Nabu Press. Paperback. Book Condition: New. This item is printed on demand. Paperback. 434 pages. Dimensions: 9.7in. x 7.4in. x 0.9in.This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781142017804

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Published by Nabu Press, United States (2013)

ISBN 10: 128753001X ISBN 13: 9781287530015

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**Item Description: **Nabu Press, United States, 2013. Paperback. Book Condition: New. Primary Source ed. 246 x 189 mm. Language: English Brand New Book ***** Print on Demand *****. This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ***** Print on Demand *****. Bookseller Inventory # APC9781287530015

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Published by Nabu Press

ISBN 10: 128753001X ISBN 13: 9781287530015

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From: BuySomeBooks (Las Vegas, NV, U.S.A.)

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**Item Description: **Nabu Press. Paperback. Book Condition: New. This item is printed on demand. Paperback. 414 pages. Dimensions: 9.7in. x 7.4in. x 0.8in.This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781287530015

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Published by Rarebooksclub.com, United States (2012)

ISBN 10: 1130655547 ISBN 13: 9781130655544

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**Item Description: **Rarebooksclub.com, United States, 2012. Paperback. Book Condition: New. 246 x 189 mm. Language: English Brand New Book ***** Print on Demand *****.This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1831 Excerpt: .Therefore the area of all the equal trapezoids which form the lateral surface of Sea note F. the frustum, is equal to the product of half the slant-height into the sum of the perimeters of the ends. Ex. If the slant-height of a frustum of a regular octagonal pyramid be 42 feet, the sides of one end 5 feet each, and the sides of the other end 3 feet each; what is the lateral surface?, Ans. 1344 square feet. 52. If the slant-height be not given, it may be obtained from the perpendicular height, and the dimensions of the two ends. Let GD (Fig. 17.) be the slant-height of the frustum CDGL, RN or GP the perpendicular height, ND and RG the radii of the circles inscribed in the perimeters of the two ends. Then PD is the difference of the two radii: And the slant-height GD = x/(GP3+YD3). Ex. If the perpendicular height of a frustum of a regular hexagonal pyramid be 24, the sides of one end 13 each, and the sides of the other end 8 each; what is the whole surface? t/(BC-BP)=CP, (Fig.7.)thati8, .v/(l33-633)=11.258 And V 83--43 e= 6.928 The difference of the two radii is, therefore, 4.33 The slant-height =V(243 + 4.333)=24.3875 The lateral surface te 1536.4 And the whole surface, 2141.75. 53. The height of the whole pyramid may be calculated from the dimensions of the frustum. Let VN (Fig. 17.) be the height of the pyramid, RN or GP the height of the frustum, ND and RG the radii of the circles inscribed in the perimeters of the ends of the frustum. Then, in the similar triangles GPD and VND, DP: GP:: DN: VN. The height of the frustum subtracted from VN, gives VR the height of the small pyramid VLG. The solidity and lateral surface of the frustum may then be found, by subtracting from the whole pyramid, the part which is above the cut ting plane. This method may serve to . Bookseller Inventory # APC9781130655544

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Published by Rarebooksclub.com, United States (2012)

ISBN 10: 1231205318 ISBN 13: 9781231205310

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**Item Description: **Rarebooksclub.com, United States, 2012. Paperback. Book Condition: New. 246 x 189 mm. Language: English Brand New Book ***** Print on Demand *****.This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1854 Excerpt: .the two ends, there may be drawn in each, such a number of small rectangles, that the sum of them shall differ less, than by any given quantity, from the figure in which they are contained. And the solids between these rectangles will be rectangular prismoids. Ex. 1. If one end of a rectangular prismoid be 44 feet by 23, the other end 36 by 21, and the perpendicular height 72; what is the solidity? The area of the larger end =44x23=1012 of the smaller end =36x21= 756 of the middle section =40x22= 880 And the solidity =(1012+756 +4x880)xl2=63456 feet. 2. What is the solidity of a stick of hewn timber, whose ends are 30 inches by 27, and 24 by 18, and whose length is 48 feet? Ans. 204 feet. Other solids not treated of in this section, if they be bounded by plane surfaces, may be measured by supposing them to be divided into prisms, pyramids, and wedges. And, in.v deed, every such sobi may be considered as made up of triangular pyramids. THE FIVE REGULAR SOLIDS. 56. A SOLID IS SAID TO BE REGULAR, WHEN ALL ITS SOLID ANGLES ARE EQUAL, AND ALL ITS SIDES ARE EQUAL AND REGULAR POLYGONS. The following figures are of this description; 1. The Tetraedron, 2. The Hexaedron or cube, 3. The Octaedron, 4. The Dodecaedron, 5. The Icosaedron, whose sides are four triangles; six squares; eight triangles; twelve pentagons; twenty triangles. Besides these five, there can be no other regular solids. The only plane figures which can form such solids, are triangles, squares, and pentagons. For the plane angles which contain any solid angle, are together. less than four right angles or 360. (Sup. Euc. 21,2.) And the least number which can form a solid angle is three. (Sup. Euc. Def. 8, 2.) If they are angles of equilateral triangles, each is 60. The sum of three of them is 180, of . Bookseller Inventory # APC9781231205310

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Published by Rarebooksclub.com, United States (2012)

ISBN 10: 1130343022 ISBN 13: 9781130343021

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**Item Description: **Rarebooksclub.com, United States, 2012. Paperback. Book Condition: New. 246 x 189 mm. Language: English Brand New Book ***** Print on Demand *****.This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1838 Excerpt: .of the base; and therefore, the sum of the breadths is equal to the perimeter of the base. Ex. 1. If the base of a right prism be a regular hexagon whose sides are each 2 feet 3 inches, and if the height be 16 feet, what is the lateral surface Ans. 216 square feet. If the areas of the two ends be added to the lateral surface, the sum will be the whole surface of the prism. And the superficies of any solid bounded by planes, is evidently equal to the areas of all its sides. Ex. 2. If the base of a prism be an equilateral triangle whose perimeter is 6 feet, and if the height be 17 feet, what is the surface 1 The area of the triangle is 1.732. (Art. 11.) And the whole surface is 105.464. PROBLEM III. To find the Solidity of a Pyramid. 48. Multiply The Area Of The Base Into i Of The Height. The solidity of a prism is equal to the product of the area of the base into the height. (Art. 43.) And a pyramid is i of a prism of the same base and altitude. (Sup. Euc. 15, 3. Cor. 1.) Therefore the solidity of a pyramid whether right or oblique, is equal to the product of the base into A of the perpendicular height. Ex. 1. What is the solidity of a triangular pyramid, whose height is 60, and each side of whose base is 41 The area of the base is 6.928 And the solidity is 138.56. 2. Let ABC (Fig. 16.) be one side of an oblique pyramid whose base is 6 feet square; let BC be 20 feet, and make an angle of 70 degrees with the plane of the base; and let CP be perpendicular to this plane. What is the.solidity of the pyramid? 3. What is the lateral surface of a regular pyramid whose axis is 12 feet, and whose base is 18 feet square? Ans. 540 square feet. The lateral surface of an oblique pyramid may be found, by taking the sum of the areas of the unequal triangles which form it. Bookseller Inventory # APC9781130343021

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ISBN 10: 114201780X ISBN 13: 9781142017804

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From: BookSleuth (Danville, CA, U.S.A.)

**Item Description: **Book Condition: Good. The Principles of Plane Trigonometry, Mensuration, Navigation and Surveying. Bookseller Inventory # Grb2496792

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